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Combinatorial applications of symmetric function theory to certain classes of permutations and truncated tableaux

ProQuest Dissertations and Theses, 2011
Dissertation
Author: Greta Cvetanova Panova
Abstract:
The purpose of this dissertation is to study certain classes of permutations and plane partitions of truncated shapes. We establish some of their enumerative and combinatorial properties. The proofs develop methods and interpretations within various fields of algebraic combinatorics, most notably the theory of symmetric functions and their combinatorial properties. The first chapter of this thesis reviews the necessary background theory on Young tableaux, symmetric functions and permutations. In the second chapter I find a bijective proof of the enumerative formula of permutations starting with a longest increasing subsequence, thereby answering a question of A.Garsia and A.Goupil. The third chapter studies the shape under the Robinson-Schensted-Knuth correspondence of separable (3142 and 2413 avoiding) permutations and proves a conjecture of G.Warrington. In the last chapter I consider Young tableaux and plane partitions of truncated shapes, i.e., whose diagrams are obtained by erasing boxes in the north-east corner from straight or shifted Young diagrams. These objects were first introduced in a special case by R.Adin and Y.Roichman and appeared to possess product-type enumerative formulas, which is a rare property shared only by a few classes of tableaux, most notably the standard Young tableaux. I find formulas for the number of standard tableaux, whose shapes are shifted staircase truncated by one box and straight rectangles truncated by a shifted staircase. The proofs involve interpretations in terms of semi-standard Young tableaux, polytope volumes, Schur function identities and the Robinson-Schensted-Knuth correspondence.

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Full document contains 76 pages
Abstract: The purpose of this dissertation is to study certain classes of permutations and plane partitions of truncated shapes. We establish some of their enumerative and combinatorial properties. The proofs develop methods and interpretations within various fields of algebraic combinatorics, most notably the theory of symmetric functions and their combinatorial properties. The first chapter of this thesis reviews the necessary background theory on Young tableaux, symmetric functions and permutations. In the second chapter I find a bijective proof of the enumerative formula of permutations starting with a longest increasing subsequence, thereby answering a question of A.Garsia and A.Goupil. The third chapter studies the shape under the Robinson-Schensted-Knuth correspondence of separable (3142 and 2413 avoiding) permutations and proves a conjecture of G.Warrington. In the last chapter I consider Young tableaux and plane partitions of truncated shapes, i.e., whose diagrams are obtained by erasing boxes in the north-east corner from straight or shifted Young diagrams. These objects were first introduced in a special case by R.Adin and Y.Roichman and appeared to possess product-type enumerative formulas, which is a rare property shared only by a few classes of tableaux, most notably the standard Young tableaux. I find formulas for the number of standard tableaux, whose shapes are shifted staircase truncated by one box and straight rectangles truncated by a shifted staircase. The proofs involve interpretations in terms of semi-standard Young tableaux, polytope volumes, Schur function identities and the Robinson-Schensted-Knuth correspondence.